# Equilibrium distributions for a finite urn scheme

Given an urn with $n$ (fixed) balls that can be red or black, and given two parameters $0 < p, \, q < 1$, keep doing the following:

1. Flip a $p$-coin. If heads come up, remove a black ball or if there are no black balls, remove a red ball. If tails come up, remove a random ball, chosen uniformly.
2. Flip a $q$-coin. If heads come up, add a black ball, otherwise add a red ball.

There is a canonical reformulation as a random walk on $\{0, 1, \dots, n\}$, where the state is the number of black balls.

I am interested in the equilibrium distribution as a function of $p, \, q$, especially in the probability that there are no black balls, for equilibrium. Any pointers will be appreciated.

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The equilibrium probability that there is no black ball is a nondecreasing function of p and a nonincreasing function of q with the obvious limits 0 or 1 when p or q goes to 0 or 1. Once these obvious remarks are made, I doubt one can expect explicit formulas (but would love to be proven wrong). – Did Nov 3 '12 at 12:58